I just finished a course on chaos and fractals, and it got me thinking about irrational numbers, and I was thinking about whether or not the distribution of the digits are chaotic.
Consider a map $\phi(n):\mathbb{Q}^c\rightarrow\mathbb{Q}^c$ defined by $n\mapsto 10\{n\}$ where $\{n\}$ represents the fractional part of $n$, so for example: $$ \phi(\pi) = 1.415926\dots, \qquad \phi^2(\pi) = 4.159265\dots $$ etc.
In any case, to prove that $\phi$ is chaotic on $\mathbb{Q}^c$ we need to show three things:
- Sensitive Dependence to Initial Conditions (SDIC): For all $x\in \mathbb{Q}^c$, and any $\delta>0$, there exists a $y\in(x-\delta,x+\delta)$ and some $k\in\mathbb{N}$ such that $\lvert\phi^k(x)-\phi^k(y)\rvert>\epsilon$ for some $\epsilon>0$.
- Transitivity: For $x,y\in \mathbb{Q}^c$ and $\delta_1,\delta_2>0$ there exists a point $p\in(x-\delta_1,x+\delta_1)$ and $k\in\mathbb{N}$ such that $\phi^k(p)\in(y-\delta_2,y+\delta_2)$.
- Regularity: The periodic points of $\phi$ are dense in $\mathbb{Q}^c$
SDIC is easy to see; pick $x\in\mathbb{Q}^c$ and pick some $\delta>0$, and a $y\in(x-\delta,x+\delta)$ we know that $x$ and $y$ have to differ at at least $1$ point in their decimal expansion, suppose they agree up to the first $k$, then $\lvert\phi^k(x)-\phi^k(y)\rvert >\epsilon$ for some $\epsilon>0$
Transitivity is satisfied because we can construct our irrational number $p$ to agree with $x$ up to so many digits say $k$ so that it is in the interval $(x-\delta_1,x+\delta_1)$, and then from the $k+1$-th digit to the $m$-th digit chosen with $y$ and $\delta_2$ in mind so that: $$ \phi^k(p) \in (y-\delta_2,y+\delta_2) $$
Lastly we come to regularity, the first problem I see is that by the nature of $x\in\mathbb{Q}^c$ having non-repeating decimal expansions $\phi^n(x)\neq x$ for any irrational number $x$, and hence $\phi$ has no $k$-periodic points, and the empty set is nowhere dense violating regularity.
It would seem that this puts a nail in the coffin of the idea. However, irrational numbers still intuitively feel chaotic to some degree. Is there a work around that I couldn't see? Or maybe some kind of pseudo-chaotic behavior that is "almost chaotic" without satisfying regularity? I would be interested in any readings on the subject. Thanks in advance!
What you are doing is already known, however for simplicity, it is usually done in binary, where it is called the the bit-shift map.